A381909 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * B(x)) ), where B(x) is the g.f. of A002293.
1, 3, 16, 121, 1117, 11569, 128648, 1500054, 18091859, 223794730, 2823369749, 36185653049, 469808971400, 6165903108879, 81667617713170, 1090234962290114, 14654059445570507, 198151602861222385, 2693625234657193038, 36789566028850640226, 504600217464088999466
Offset: 0
Keywords
Programs
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PARI
a(n) = sum(k=0, n, binomial(n+4*k+1, k)*binomial(2*n+2, n-k)/(n+4*k+1));
Formula
G.f. A(x) satisfies A(x) = (1 + x*A(x))^2 * B(x*A(x)).
a(n) = Sum_{k=0..n} binomial(n+4*k+1,k) * binomial(2*n+2,n-k)/(n+4*k+1).
a(n) = binomial(2*(1 + n), n)*hypergeom([(1+n)/4, (2+n)/4, (3+n)/4, (4+n)/4, -n], [(2+n)/3, (3+n)/3, (4+n)/3, 3+n], -2^8/3^3)/(1 + n). - Stefano Spezia, Mar 10 2025